"Lea is deeply grateful for all the love and support she’s received from family, friends, and fans. Since Cory’s passing, Lea has been grieving alongside his family and making appropriate arrangements with them," reads the statement, obtained by PEOPLE exclusively.

"They are supporting each other as they endure this profound loss together. We continue to ask the media to respect the privacy of Lea and Cory’s family.”

Monteith, 31, whose body was found in a Vancouver hotel room over the weekend, died in what the coroner called a "most-tragic accident."